FUZ: perform fuzzy logic fuzzification and defuzzification (H)

• Fuzzy control/decisions based on empirical rules may be an interesting alternative when a mathematical model does not exist or when processing is too complex. In combination with the COM function for serial communication it is easy to develop effective control systems.
• • FUZ( [overlap,] FuzzySet, LinguisticVariables ) = input_value ! fuzzification
  • output_value = FUZ( [overlap,] FuzzySet, LinguisticVariables) ! defuzzification
• FuzzySet is a vector of fuzzy memberships like
  • REAL :: prices(3) ! define the vector prices with 3 elements
  • prices = (1, 5, 20) ! give prices specific values
• LinguisticVariables is either a vector or a series of scalars with the same number of elements as FuzzySet to hold the partial membership of a specific value to FuzzySet like
  • linguistic_variables = ( cheap, average, expensive )
• build a

  fuzzy expert system

  in 3 levels:

  1. fuzzification

     : obtain fuzzy set memberships of inputs by
     • FUZ(set_i, vars_i) = input_i
     example:
     • FUZ(prices, cheap, average, expensive) = 10 ! sets cheap=0, average=2/3, expensive=1/3
     • FUZ(qualities, q_vector) = this_sample ! sets the elements of q_vector

  2. inferences

     : make rules to deduce linguistic output memberships from linguistic input memberships like
     • good_value = MIN(cheap, high_quality , pretty) ! MIN as the fuzzy operator AND
     • low_value = MAX(expensive, low_qualitity) ! MAX as fuzzy operator OR
     • no_value = (long_delivery + ugly)/2 ! no reason not to invent your own fuzzy operators, fuzzy is very docile!

  3. defuzzification

     : combine inference variables to a specific output by
     • output = FUZ(set_o, vars_o)
     example:
     • rating = FUZ(scale, no_value, low_value, good_value) ! calculated as memberships "center of gravity" in scale
• overlap is an optional number of neighbors in FuzzySet (default = 1):
  • overlap = 1 (default): values are distributed to 1 left and 1 right member
  • FUZ(n, FuzzySet, LingVars) extends this to n neighbors. This can help to smooth the results.
• linguistic variables need not be normalized
• best results seem to be obtained for odd dimensions of the fuzzy sets
• FuzzySets are handled internally as triangular distributions in HicEst

• Demonstration of a simple fuzzy braking algorithm

  1. fuzzification

     (distribute input to linguistic variables):

      REAL ::  REAL :: speed_set(3), distance_set(3), brake_set(3)
     
           speed_set = ( 0,   50,  100)         ! km/h
      FUZ( speed_set, slow, town, fast) = speed 
      ! eg speed=40:  0.2,  0.8,  0 
     
           distance_set = (  5,     30,   100)  ! m
      FUZ( distance_set, short, medium,  long)  = distance 
      ! eg distance=65;  0,     0.5,     0.5

  2. inferences

     :

      soft     = MAX(slow, long)   ! eg 0.5
      moderate = MAX(town, medium) ! eg 0.8
      hard     = MAX(fast, short)  ! eg 0

  3. defuzzification

     :

                  brake_set = ( 0,      0.3,    1)
      brake = FUZ(brake_set, soft, moderate, hard)
     

  4. graphical test

     :
     • note the nonlinearities even for this extremely simple model
     • results become smoother with overlap > 1 and with more inferences

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